Results in descriptive set theory on some represented spaces

Abstract

Descriptive set theory was originally developed on Polish spaces. It was later extended to $\omega$-continuous domains [Selivanov 2004] and recently to quasi-Polish spaces [de Brecht 2013]. All these spaces are countably-based. Extending descriptive set theory and its effective counterpart to general represented spaces, including non-countably-based spaces has been started in [Pauly, de Brecht 2015]. We study the spaces $\mathcal{O}({\mathbb{N}}^{\mathbb{N}})$, $\mathcal{C}({\mathbb{N}}^{\mathbb{N}},2)$ and the Kleene-Kreisel spaces $\mathbb{N}⟨\alpha ⟩$. We show that there is a ${\Sigma }_2^0$-subset of $\mathcal{O}({\mathbb{N}}^{\mathbb{N}})$ which is not Borel. We show that the open subsets of ${\mathbb{N}}^{{\mathbb{N}}^{\mathbb{N}}}$ cannot be continuously indexed by elements of ${\mathbb{N}}^{\mathbb{N}}$ or even ${\mathbb{N}}^{{\mathbb{N}}^{\mathbb{N}}}$, and more generally that the open subsets of $\mathbb{N}⟨\alpha ⟩$ cannot be continuously indexed by elements of $\mathbb{N}⟨\alpha ⟩$. We also derive effective versions of these results. These results give answers to recent open questions on the classification of spaces in terms of their base-complexity, introduced in [de Brecht, Schröder, Selivanov 2016]. In order to obtain these results, we develop general techniques which are refinements of Cantor's diagonal argument involving multi-valued fixed-point free functions and that are interesting on their own right.